919 Words4 Pages

Beyond Pythagoras
What this coursework has asked me to do is to investigate and find a
generalisation, for a family of Pythagorean triples. This will include
odd numbers and even numbers.
I am going to investigate a family of right-angled triangles for which
all the lengths are positive integers and the shortest is an odd
number.
I am going to check that the Pythagorean triples (5,12,13) and
(7,24,25) cases work; and then spot a connection between the middle
and longest sides.
The first case of a Pythagorean triple I will look at is:
[IMAGE]
[IMAGE][IMAGE][IMAGE]The numbers 5, 12 and 13 satisfy the connection.
5² + 12² = 13²
25 + 144 = 169
169 = 13
The second case of a Pythagorean triple I will look at is:
[IMAGE][IMAGE]The numbers 7, 24 and 25 satisfy the connection.
7² + 24² = 25²
[IMAGE][IMAGE][IMAGE]49 + 576 = 625
625 = 25
There is a connection between the middle and longest side. This is
that there is a one number difference.
So if M= middle and L= longest
L = M + 1
I am going to use the triples, (3,4,5), (5,12,13) and (7,24,25) to
find other triples. Then I will put my results in a table and look for
a pattern that will occur. I will then try and predict the next
results in the table and prove it.
[IMAGE]
[IMAGE]
[IMAGE]
n
Smallest
O
Middle
O
Longest
O
1
3
4
5
2
5
12
13
3
7
24
25
There is a clear pattern between the middle and longest side.
There is also a sequence forming.
n = 1 S = 3 M = 4 L = 5
n = 2 S = 5 M = 12 L = 13

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